Let’s practice finding the focus of the parabola. A parabola graphed using the equation y=ax2 must first be converted into standard form. Standard form for parabolas is as follows:
4p(y - k) = (x - h)2
Where h is the horizontal distance of the parabola origin (the curve) of the parabola from the point (0,0) and k is the vertical distance of the parabola origin from the point (0,0). 4p is equal to (1/a). This is just a mathematics convention.
We plug our numbers in to solve for p, which is the vertical distance of the focus from the parabola’s vertex, or lowest point.
So why would we need to use a parabola as our mirror, anyway? Here’s what’s so cool about parabolic mirrors: the focus is the point where all of the reflected light passes through. This makes a parabola a perfect mirror shape for cooking a hot dog.
Build a solar power hot dog cooker.
- Oversized shoe box
- Aluminum foil
- Poster board
- Craft knife or box cutter
- Single hole punch
- Sheet of graphing paper
- Hot dogs, buns, and your favorite hot dog condiments!
- Using the graphing paper and a pencil, graph the parabola y = 0.035x2. (Because this parabola is facing up, you can put the vertex of the parabola very close to the bottom of the page. Be sure to scale the parabola appropriately to the graph paper; for example, if the squares on the paper are ¼ inch, make 4 boxes equal 1 inch, and mark your axes accordingly. Follow the same steps if you're using centimeters.)
- Cut the along the parabola line so you have a template of the curve.
- On a separate sheet of paper, calculate the focus of the parabola. Remember—why is this important?
- Remove the lid to the shoebox. Trace the parabola curve on each long side of the shoe box.
- Have an adult help you cut out the shape of the parabola on the box.
- Use the scissors to cut out a piece of poster board that fits perfectly into the curve of the parabola in the box.
- Glue aluminum foil to the poster board, shiny side up. Try to keep the foil as smooth as possible.
- Carefully insert the foil-covered poster board into the shoebox and tape it in place. Be sure the shiny side of the foil faces you.
- Use two long scraps of cardboard to build supports for the skewer. They must be tall enough to hold the skewer above the focus.
- Use the single hole punch to make holes at the focal point, which will be at the same height as the focus is from the vertex of your parabola. If you aren’t sure where it is, take your cooker outside, tilt it towards the sun, and look at where the sun shines brightest on the inside of your cardboard supports.
- Attach a hot dog to a long skewer and set the skewer in the support holes to keep the hot dog suspended above the parabola. Direct the parabola towards the sun by propping one side up slightly.
- Turn the hot dog every so often to cook it evenly. Enjoy!
The focus of the parabola is located at the coordinates (0, 7.14), which is 7.14 units (inches or centimeters, depending on the measurement you've chosen to use) above the vertex (bottom) of the parabola’s curve.
Placing the hot dog at the focus in the sun will result in a cooked (and ready to eat!) hot dog.
The parabola is shaped so that it collects the sun rays and focuses them at one point, the focus, in the center of the parabola. This is where the hot dog is placed, and the energy from the sun is used to cook the hot dog.
Because the sun is so far away from the earth, the light rays hitting us are essentially parallel. Parallel incident rays of light which strike a parabolic mirror all pass through the same point after they are reflected.
Light is reflected off of nearly everything, but we used foil because it’s highly reflective and much of the heat and energy from the incoming sun rays is redirected to the hot dog. This would not work with a material that was not reflective.